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                    <h1 class="description center-align post-title">洛谷UVA00184题解</h1>
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                <h3 id="一、翻译勘误"><a href="#一、翻译勘误" class="headerlink" title="一、翻译勘误"></a>一、翻译勘误</h3><p>题目翻译中写到：每个测试用例最多有 $100$ 个节点。</p>
<p>但是有一个测试用例是例外。这个测试用例亲测应该少于 $1000$ 个节点（<a target="_blank" rel="noopener" href="https://www.luogu.com.cn/user/387840">dengziyue</a> 的说法：少于 $300$ 个）。</p>
<p>作为本题的翻译者，在此向大家谢罪。</p>
<h3 id="二、前置芝士：如何判断三个点处于同一条直线"><a href="#二、前置芝士：如何判断三个点处于同一条直线" class="headerlink" title="二、前置芝士：如何判断三个点处于同一条直线"></a>二、前置芝士：如何判断三个点处于同一条直线</h3><p>设三个点：$A(x_1,y_1)$，$B(x_2,y_2)$，$C(x_3,y_3)$，并且这三个点不位于同一位置。</p>
<p>当 $(y_2-y_1)\div(x_2-x_1)=(y_3-y_2)\div(x_3-x_2)$ 时，点 $A$、点 $B$、点 $C$ 位于同一条直线上。</p>
<p>但是有一个前提：$x_1\ne x_2$ 且 $x_2\ne x_3$。</p>
<p>因此我们增加特判：</p>
<ol>
<li><p>如果 $x_1=x_2$，则：如果 $x_2=x_3$，则三点处于同一条直线，否则就不处于同一条直线。</p>
</li>
<li><p>如果 $x_2=x_3$，则：如果 $x_1=x_2$，则三点处于同一条直线，否则就不处于同一条直线。</p>
</li>
</ol>
<p>对于纵坐标，逻辑也相同。</p>
<p>因此我们得出了判断三个点是否处于同一条直线的代码：</p>
<pre class="line-numbers language-none"><code class="language-none">bool check_point_in_line(int x1,int y1,int x2,int y2,int x3,int y3)
&#123;
	if(x1&#x3D;&#x3D;x2)
	&#123;
		return x2&#x3D;&#x3D;x3;
	&#125;
	if(x2&#x3D;&#x3D;x3)
	&#123;
		return x1&#x3D;&#x3D;x2;
	&#125;
	if(y1&#x3D;&#x3D;y2)
	&#123;
		return y2&#x3D;&#x3D;y3;
	&#125;
	if(y2&#x3D;&#x3D;y3)
	&#123;
		return y1&#x3D;&#x3D;y2;
	&#125;
	return (double)(y2-y1)&#x2F;(x2-x1)&#x3D;&#x3D;(double)(y3-y2)&#x2F;(x3-x2);
&#125;<span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<h3 id="三、具体思路"><a href="#三、具体思路" class="headerlink" title="三、具体思路"></a>三、具体思路</h3><p>首先读入数据，并且按照题目要求进行排序。</p>
<p>然后，枚举两个节点 $a$ 和 $b$，作为直线的其中两个点。</p>
<p>接着，枚举几个节点，如果枚举到的节点与 $a$ 和 $b$ 处于同一条直线上，那么加入这条直线的节点集。</p>
<p>最后，如果检测到这条直线的节点集只有 $a$ 和 $b$，则排除这条直线。</p>
<p>使用一个三级循环即可解决问题。</p>
<p>但是，我们需要一个关键步骤：去重。</p>
<p>假设这是某一个测试用例的输入：</p>
<pre class="line-numbers language-none"><code class="language-none">1 1 2 2 3 3 4 4 0 0<span aria-hidden="true" class="line-numbers-rows"><span></span></span></code></pre>
<p>显然，只有一条直线，所以这是正确输出：</p>
<pre class="line-numbers language-none"><code class="language-none">The following lines were found:
(   1,   1)(   2,   2)(   3,   3)(   4,   4)
<span aria-hidden="true" class="line-numbers-rows"><span></span><span></span></span></code></pre>
<p>但实际情况是，输出是这样的：</p>
<pre class="line-numbers language-none"><code class="language-none">The following lines were found:
(   1,   1)(   2,   2)(   3,   3)(   4,   4)
(   1,   1)(   3,   3)(   4,   4)
(   2,   2)(   3,   3)(   4,   4)
<span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span></span></code></pre>
<p>可以看到，多出来了两条直线，因为输出的第一条直线就包含了第二条直线和第三条直线。</p>
<p>那么如何去重？</p>
<h3 id="四、关键步骤：去重"><a href="#四、关键步骤：去重" class="headerlink" title="四、关键步骤：去重"></a>四、关键步骤：去重</h3><p>首先，将输入的节点排序后，每个节点都赋予一个唯一编号：在输入存储数组的下标。</p>
<p>然后，上述错误输出对应的唯一编号就是：</p>
<pre class="line-numbers language-none"><code class="language-none">1 2 3 4
1 3 4
2 3 4<span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span></span></code></pre>
<p>我们可以发现，第二条直线有的编号，第一条直线都有。同理，第三条直线有的编号，第一条直线都有。</p>
<p>此时，我们说这三条直线“互相重合”。</p>
<p>由于循环顺序的特殊性，如果几条直线互相重合，第一条直线包含的节点数量一定大于之后的直线的节点数量。所以对于互相重合的直线，我们只需保留第一个即可，不必纠结那条直线才是正确答案。</p>
<p>而 $n\le300$，可以使用桶的思想去做。</p>
<p>因此我们得出了去重代码：</p>
<pre class="line-numbers language-cpp" data-language="cpp"><code class="language-cpp"><span class="token keyword">bool</span> <span class="token function">check</span><span class="token punctuation">(</span><span class="token keyword">int</span> line_id<span class="token punctuation">)</span><span class="token comment">//line_id代表：第line_id条直线是否可以与其它直线共同存在（不重合）？重合返回false，不重合返回true。</span>
<span class="token punctuation">&#123;</span>
	<span class="token keyword">bool</span> check_appear<span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">;</span><span class="token comment">//第line_id条直线包含的编号。</span>
	<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>i<span class="token operator">&lt;</span>MAXN<span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		check_appear<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">false</span><span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>i<span class="token operator">&lt;=</span>main_point_cnt<span class="token punctuation">[</span>line_id<span class="token punctuation">]</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">int</span> n_index<span class="token operator">=</span>line<span class="token punctuation">[</span>line_id<span class="token punctuation">]</span><span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>
		check_appear<span class="token punctuation">[</span>n_index<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">true</span><span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>i<span class="token operator">&lt;</span>line_id<span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">bool</span> flag<span class="token operator">=</span><span class="token boolean">false</span><span class="token punctuation">;</span><span class="token comment">//如果出现了其它直线不包含但第line_id条直线包含的点，那么肯定不重合。</span>
		<span class="token keyword">bool</span> appear<span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">;</span><span class="token comment">//其它直线包含的编号。</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>j<span class="token operator">&lt;</span>MAXN<span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			appear<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">false</span><span class="token punctuation">;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>j<span class="token operator">&lt;=</span>main_point_cnt<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">int</span> n_index<span class="token operator">=</span>line<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
			appear<span class="token punctuation">[</span>n_index<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">true</span><span class="token punctuation">;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>j<span class="token operator">&lt;</span>MAXN<span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">if</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token operator">!</span>appear<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token operator">&amp;&amp;</span>check_appear<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">)</span>
			<span class="token punctuation">&#123;</span>
				flag<span class="token operator">=</span><span class="token boolean">true</span><span class="token punctuation">;</span>
				<span class="token keyword">break</span><span class="token punctuation">;</span>
			<span class="token punctuation">&#125;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">if</span><span class="token punctuation">(</span>flag<span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">continue</span><span class="token punctuation">;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">return</span> <span class="token boolean">false</span><span class="token punctuation">;</span><span class="token comment">//出现了重合。</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">return</span> <span class="token boolean">true</span><span class="token punctuation">;</span><span class="token comment">//未出现重合。</span>
<span class="token punctuation">&#125;</span><span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<h3 id="五、完整代码"><a href="#五、完整代码" class="headerlink" title="五、完整代码"></a>五、完整代码</h3><pre class="line-numbers language-cpp" data-language="cpp"><code class="language-cpp"><span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span><span class="token string">&lt;cstdio></span></span>
<span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span><span class="token string">&lt;algorithm></span></span>
<span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">include</span><span class="token string">&lt;cstring></span></span>
<span class="token keyword">using</span> <span class="token keyword">namespace</span> std<span class="token punctuation">;</span>
<span class="token macro property"><span class="token directive-hash">#</span><span class="token directive keyword">define</span> <span class="token macro-name">MAXN</span> <span class="token expression"><span class="token number">1005</span></span></span>
<span class="token keyword">struct</span> <span class="token class-name">point</span><span class="token comment">//节点结构体。sort不支持以数组为单位进行排序。</span>
<span class="token punctuation">&#123;</span>
	<span class="token keyword">int</span> x<span class="token punctuation">,</span>y<span class="token punctuation">;</span>
<span class="token punctuation">&#125;</span>input<span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">int</span> line<span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">;</span><span class="token comment">//存储答案直线。</span>
<span class="token keyword">int</span> main_point_cnt<span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">;</span><span class="token comment">//存储一条直线有多少个节点。</span>
<span class="token keyword">bool</span> <span class="token function">point_cmp</span><span class="token punctuation">(</span>point a<span class="token punctuation">,</span>point b<span class="token punctuation">)</span><span class="token comment">//sort的比较函数。</span>
<span class="token punctuation">&#123;</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>a<span class="token punctuation">.</span>x<span class="token operator">==</span>b<span class="token punctuation">.</span>x<span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">return</span> a<span class="token punctuation">.</span>y<span class="token operator">&lt;</span>b<span class="token punctuation">.</span>y<span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">return</span> a<span class="token punctuation">.</span>x<span class="token operator">&lt;</span>b<span class="token punctuation">.</span>x<span class="token punctuation">;</span>
<span class="token punctuation">&#125;</span>
<span class="token keyword">bool</span> <span class="token function">check</span><span class="token punctuation">(</span><span class="token keyword">int</span> line_id<span class="token punctuation">)</span>
<span class="token punctuation">&#123;</span>
	<span class="token comment">//查询是否有重复</span>
	<span class="token keyword">bool</span> check_appear<span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">;</span>
	<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>i<span class="token operator">&lt;</span>MAXN<span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		check_appear<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">false</span><span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>i<span class="token operator">&lt;=</span>main_point_cnt<span class="token punctuation">[</span>line_id<span class="token punctuation">]</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">int</span> n_index<span class="token operator">=</span>line<span class="token punctuation">[</span>line_id<span class="token punctuation">]</span><span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>
		check_appear<span class="token punctuation">[</span>n_index<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">true</span><span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>i<span class="token operator">&lt;</span>line_id<span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">bool</span> flag<span class="token operator">=</span><span class="token boolean">false</span><span class="token punctuation">;</span>
		<span class="token keyword">bool</span> appear<span class="token punctuation">[</span>MAXN<span class="token punctuation">]</span><span class="token punctuation">;</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>j<span class="token operator">&lt;</span>MAXN<span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			appear<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">false</span><span class="token punctuation">;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>j<span class="token operator">&lt;=</span>main_point_cnt<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">int</span> n_index<span class="token operator">=</span>line<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
			appear<span class="token punctuation">[</span>n_index<span class="token punctuation">]</span><span class="token operator">=</span><span class="token boolean">true</span><span class="token punctuation">;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>j<span class="token operator">&lt;</span>MAXN<span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">if</span><span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token operator">!</span>appear<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token operator">&amp;&amp;</span>check_appear<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">)</span>
			<span class="token punctuation">&#123;</span>
				flag<span class="token operator">=</span><span class="token boolean">true</span><span class="token punctuation">;</span>
				<span class="token keyword">break</span><span class="token punctuation">;</span>
			<span class="token punctuation">&#125;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">if</span><span class="token punctuation">(</span>flag<span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">continue</span><span class="token punctuation">;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token keyword">return</span> <span class="token boolean">false</span><span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">return</span> <span class="token boolean">true</span><span class="token punctuation">;</span>
<span class="token punctuation">&#125;</span>
<span class="token keyword">bool</span> <span class="token function">check_point_in_line</span><span class="token punctuation">(</span><span class="token keyword">int</span> x1<span class="token punctuation">,</span><span class="token keyword">int</span> y1<span class="token punctuation">,</span><span class="token keyword">int</span> x2<span class="token punctuation">,</span><span class="token keyword">int</span> y2<span class="token punctuation">,</span><span class="token keyword">int</span> x3<span class="token punctuation">,</span><span class="token keyword">int</span> y3<span class="token punctuation">)</span>
<span class="token punctuation">&#123;</span>
	<span class="token comment">//查询三个点是否在同一条直线上。</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>x1<span class="token operator">==</span>x2<span class="token punctuation">)</span><span class="token comment">//特判。</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">return</span> x2<span class="token operator">==</span>x3<span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>x2<span class="token operator">==</span>x3<span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">return</span> x1<span class="token operator">==</span>x2<span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>y1<span class="token operator">==</span>y2<span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">return</span> y2<span class="token operator">==</span>y3<span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">if</span><span class="token punctuation">(</span>y2<span class="token operator">==</span>y3<span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token keyword">return</span> y1<span class="token operator">==</span>y2<span class="token punctuation">;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">return</span> <span class="token punctuation">(</span><span class="token keyword">double</span><span class="token punctuation">)</span><span class="token punctuation">(</span>y2<span class="token operator">-</span>y1<span class="token punctuation">)</span><span class="token operator">/</span><span class="token punctuation">(</span>x2<span class="token operator">-</span>x1<span class="token punctuation">)</span><span class="token operator">==</span><span class="token punctuation">(</span><span class="token keyword">double</span><span class="token punctuation">)</span><span class="token punctuation">(</span>y3<span class="token operator">-</span>y2<span class="token punctuation">)</span><span class="token operator">/</span><span class="token punctuation">(</span>x3<span class="token operator">-</span>x2<span class="token punctuation">)</span><span class="token punctuation">;</span><span class="token comment">//返回计算结果。虽然double会有误差，但基本可以忽略。</span>
<span class="token punctuation">&#125;</span>
<span class="token keyword">int</span> <span class="token function">main</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
<span class="token punctuation">&#123;</span>
	<span class="token keyword">while</span><span class="token punctuation">(</span><span class="token boolean">true</span><span class="token punctuation">)</span>
	<span class="token punctuation">&#123;</span>
		<span class="token comment">//输入阶段</span>
		<span class="token keyword">int</span> point_cnt<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span><span class="token comment">//输入了多少个点。</span>
		<span class="token keyword">while</span><span class="token punctuation">(</span><span class="token boolean">true</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			point_cnt<span class="token operator">++</span><span class="token punctuation">;</span>
			<span class="token function">scanf</span><span class="token punctuation">(</span><span class="token string">"%d%d"</span><span class="token punctuation">,</span><span class="token operator">&amp;</span>input<span class="token punctuation">[</span>point_cnt<span class="token punctuation">]</span><span class="token punctuation">.</span>x<span class="token punctuation">,</span><span class="token operator">&amp;</span>input<span class="token punctuation">[</span>point_cnt<span class="token punctuation">]</span><span class="token punctuation">.</span>y<span class="token punctuation">)</span><span class="token punctuation">;</span>
			<span class="token keyword">if</span><span class="token punctuation">(</span>input<span class="token punctuation">[</span>point_cnt<span class="token punctuation">]</span><span class="token punctuation">.</span>x<span class="token operator">==</span><span class="token number">0</span><span class="token operator">&amp;&amp;</span>input<span class="token punctuation">[</span>point_cnt<span class="token punctuation">]</span><span class="token punctuation">.</span>y<span class="token operator">==</span><span class="token number">0</span><span class="token punctuation">)</span><span class="token comment">//如果输入为两个0：</span>
			<span class="token punctuation">&#123;</span>
				<span class="token keyword">if</span><span class="token punctuation">(</span>point_cnt<span class="token operator">==</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token comment">//如果还没有产生任何有效输入：</span>
				<span class="token punctuation">&#123;</span>
					<span class="token keyword">return</span> <span class="token number">0</span><span class="token punctuation">;</span><span class="token comment">//直接退出程序，完结撒花！</span>
				<span class="token punctuation">&#125;</span>
				<span class="token keyword">break</span><span class="token punctuation">;</span><span class="token comment">//退出输入循环。</span>
			<span class="token punctuation">&#125;</span>
		<span class="token punctuation">&#125;</span>
		point_cnt<span class="token operator">--</span><span class="token punctuation">;</span>
		<span class="token function">sort</span><span class="token punctuation">(</span>input<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">,</span>input<span class="token operator">+</span>point_cnt<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">,</span>point_cmp<span class="token punctuation">)</span><span class="token punctuation">;</span><span class="token comment">//排序。</span>
		<span class="token comment">//计算阶段</span>
		<span class="token keyword">int</span> line_cnt<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span><span class="token comment">//直线数量。</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>i<span class="token operator">&lt;=</span>point_cnt<span class="token operator">-</span><span class="token number">2</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span>i<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">;</span>j<span class="token operator">&lt;=</span>point_cnt<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span><span class="token comment">//枚举两个点，即前面提到过的a和b。</span>
			<span class="token punctuation">&#123;</span>
				<span class="token keyword">int</span> line_point_cnt<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span><span class="token comment">//直线点数。</span>
				line_cnt<span class="token operator">++</span><span class="token punctuation">;</span>
				<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> k<span class="token operator">=</span>j<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">;</span>k<span class="token operator">&lt;=</span>point_cnt<span class="token punctuation">;</span>k<span class="token operator">++</span><span class="token punctuation">)</span><span class="token comment">//枚举第三个点。</span>
				<span class="token punctuation">&#123;</span>
					<span class="token keyword">if</span><span class="token punctuation">(</span><span class="token function">check_point_in_line</span><span class="token punctuation">(</span>input<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">.</span>x<span class="token punctuation">,</span>input<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">.</span>y<span class="token punctuation">,</span>input<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">.</span>x<span class="token punctuation">,</span>input<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">.</span>y<span class="token punctuation">,</span>input<span class="token punctuation">[</span>k<span class="token punctuation">]</span><span class="token punctuation">.</span>x<span class="token punctuation">,</span>input<span class="token punctuation">[</span>k<span class="token punctuation">]</span><span class="token punctuation">.</span>y<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token comment">//三个点在一条直线上：</span>
					<span class="token punctuation">&#123;</span>
						<span class="token keyword">if</span><span class="token punctuation">(</span>line_point_cnt<span class="token operator">==</span><span class="token number">0</span><span class="token punctuation">)</span><span class="token comment">//如果这是找到的第一个符合要求的“第三个点”，则将前两个点加入节点集合。</span>
						<span class="token punctuation">&#123;</span>
							line<span class="token punctuation">[</span>line_cnt<span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token operator">=</span>i<span class="token punctuation">;</span>
							line<span class="token punctuation">[</span>line_cnt<span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">2</span><span class="token punctuation">]</span><span class="token operator">=</span>j<span class="token punctuation">;</span>
							line_point_cnt<span class="token operator">=</span><span class="token number">2</span><span class="token punctuation">;</span>
						<span class="token punctuation">&#125;</span>
						line_point_cnt<span class="token operator">++</span><span class="token punctuation">;</span>
						line<span class="token punctuation">[</span>line_cnt<span class="token punctuation">]</span><span class="token punctuation">[</span>line_point_cnt<span class="token punctuation">]</span><span class="token operator">=</span>k<span class="token punctuation">;</span>
					<span class="token punctuation">&#125;</span>
				<span class="token punctuation">&#125;</span>
				main_point_cnt<span class="token punctuation">[</span>line_cnt<span class="token punctuation">]</span><span class="token operator">=</span>line_point_cnt<span class="token punctuation">;</span><span class="token comment">//存储一条直线上有多少节点。</span>
				<span class="token keyword">if</span><span class="token punctuation">(</span>line_point_cnt<span class="token operator">==</span><span class="token number">0</span><span class="token punctuation">)</span><span class="token comment">//没有节点，无效。</span>
				<span class="token punctuation">&#123;</span>
					line_cnt<span class="token operator">--</span><span class="token punctuation">;</span>
				<span class="token punctuation">&#125;</span>
				<span class="token keyword">else</span> <span class="token keyword">if</span><span class="token punctuation">(</span><span class="token operator">!</span><span class="token function">check</span><span class="token punctuation">(</span>line_cnt<span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token comment">//重合，无效。</span>
				<span class="token punctuation">&#123;</span>
					line_cnt<span class="token operator">--</span><span class="token punctuation">;</span>
				<span class="token punctuation">&#125;</span>
			<span class="token punctuation">&#125;</span>
		<span class="token punctuation">&#125;</span>
		<span class="token comment">//输出阶段</span>
		<span class="token keyword">if</span><span class="token punctuation">(</span>line_cnt<span class="token operator">==</span><span class="token number">0</span><span class="token punctuation">)</span><span class="token comment">//当没有直线时：</span>
		<span class="token punctuation">&#123;</span>
			<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"No lines were found\n"</span><span class="token punctuation">)</span><span class="token punctuation">;</span><span class="token comment">//输出无直线信息。</span>
			<span class="token keyword">continue</span><span class="token punctuation">;</span><span class="token comment">//继续下一轮测试样例。</span>
		<span class="token punctuation">&#125;</span>
		<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"The following lines were found:\n"</span><span class="token punctuation">)</span><span class="token punctuation">;</span><span class="token comment">//先输出开头。</span>
		<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> i<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>i<span class="token operator">&lt;=</span>line_cnt<span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
		<span class="token punctuation">&#123;</span>
			<span class="token keyword">for</span><span class="token punctuation">(</span><span class="token keyword">int</span> j<span class="token operator">=</span><span class="token number">1</span><span class="token punctuation">;</span>j<span class="token operator">&lt;=</span>main_point_cnt<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>j<span class="token operator">++</span><span class="token punctuation">)</span>
			<span class="token punctuation">&#123;</span>
				<span class="token keyword">int</span> x<span class="token operator">=</span>input<span class="token punctuation">[</span>line<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">]</span><span class="token punctuation">.</span>x<span class="token punctuation">,</span>y<span class="token operator">=</span>input<span class="token punctuation">[</span>line<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">]</span><span class="token punctuation">.</span>y<span class="token punctuation">;</span>
				<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"(%4d,%4d)"</span><span class="token punctuation">,</span>x<span class="token punctuation">,</span>y<span class="token punctuation">)</span><span class="token punctuation">;</span><span class="token comment">//输出每个节宽点。注意输出场宽。</span>
			<span class="token punctuation">&#125;</span>
			<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"\n"</span><span class="token punctuation">)</span><span class="token punctuation">;</span><span class="token comment">//换行。</span>
		<span class="token punctuation">&#125;</span>
	<span class="token punctuation">&#125;</span>
	<span class="token keyword">return</span> <span class="token number">0</span><span class="token punctuation">;</span><span class="token comment">//虽然程序结构决定了程序不可能执行到这里，但还是加一句。</span>
<span class="token punctuation">&#125;</span>
<span aria-hidden="true" class="line-numbers-rows"><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span><span></span></span></code></pre>
<h3 id="六、后记"><a href="#六、后记" class="headerlink" title="六、后记"></a>六、后记</h3><p>这是我到目前为止写过的最长的一篇题解，因为这道题实在太难。</p>
<p>记录：<a target="_blank" rel="noopener" href="https://www.luogu.com.cn/record/51815697">https://www.luogu.com.cn/record/51815697</a>。</p>
<p>By <a target="_blank" rel="noopener" href="https://www.luogu.com.cn/user/387836">dengzijun</a></p>

                
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